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Table V Characteristics of the a Relaxation Sample
T,,(E”)a
T,,(tan 6)“
HYC3h HYC4 HYC5h HYC6 HYC7 HYC8
-20
-8
-12
8 20 63 70
” At 110 Hz.
a relaxation and
0
30 76 118 127
tan 6 magnitude“ 0.45 0.46 0.50 0.13
0.16 0.18
p relaxation overlap.
the great influence of the crystal phase in polyethylene-like materials on this motion. The a Relaxation. The characteristics of the a relaxation are collected in Table V. The overlap of the a and /3 relaxations in samples HYCS, HYC4, and HYC5 makes a quantitative discussion difficult. However, there seems little doubt that in these derivatives of very low crystallinity, the a relaxation originates in motions accompanying the melting of small imperfect crystals and not from the conventional polyethylene a mechanism involving intracrystalline chain motions prior to melting. Table V indicates that the melting range in these derivatives encompasses the temperature of the a relaxation. On the other hand, the a relaxation in samples HYC6, HYC7, and HYC8 very probably does originate in intracrystalline motions and perhaps partially in interlamellar slip mechanisms as well.
Conclusions The series of studies on the polypentenamer derivatives together with those of other workers on a number of other polymers begin to point toward the general conclusion that the two-phase model is an adequate representation of polymer behavior only in the case where the “amorphous phase” is of
a sufficiently different structure from the “crystalline phase” that it cannot crystallize under any circumstances. Thus in isotactic polystyrene, crystallinity has no effect on the glass transition temperature or relaxations accompanying it and the same is true of polypropylene (and the cis-trans isomers of polypentenamer). On the other hand, polyethylene terephthalate and the polypentenamer derivatives have glass transition behavior which is greatly affected by crystallinity. The mechanism of the interaction of the “crystalline” and “amorphous” phases is by no means clear. The elucidation of the nature of this interaction will require extensive additional investigation.
Acknowledgments. The authors are grateful to the National Science Foundation under Grant DMR 75-06916 for partial support of this research. The use of NSF Materials Research Laboratory facilities is acknowledged. We are also grateful to Dr. N. Calderon of the Goodyear Tire and Rubber Co. for providing the polypentenamer samples and to Dr. S. Matsuoka of Bell Laboratories for helpful discussions. References and Notes (1) K. Sanui, W. J. MacKnight, and R. W. Lenz, Macromolecules, 7, 101 (1974). ( 2 ) K. H. Illers, Kolloid Z. 2. Polym., 250,426 (1972). (3) C. E. Wilkes, M. J. P. Peklo, and R. J. Minchak, J . Polym. Sci.,Part C , 43, 97 (1973). (4) G . Dall’Asta and P. Scaglione, Rubber Chem. Techno/..42,1235 (1969). ( 5 ) K . Sanui, W. d. MacKnight, and R. W. Lenz, J . Polym. Sci., Part R, 11, 427 (1973). ( 6 ) C . Tosi, F. Ciampelli, and G. Dall’Asta, J . Polym. Sci., Poiym. Phys. Ed., 11,529(1973). ( 7 ) B. E. Read and R. S.Stein, Macromolecules, 1,116 (1968). ( 8 ) J . D. Hoffman, G. Williams, and E. Passaglia, J . Polqrn. Sci., Part C , 15, 10 (1966). (9) N. G. McCrum, B. E. Read, and G. Williams, “Anelastic and Dielectric Effects in Polymeric Solids”, Wiley, New York, N.Y., 1967, Chapter 10. (10) R. H. Boyd and S. M. Breitling, Macromolecdes. 7,855 (1974). (11) d. M. Crissman and E. Passaglia, J Appl. Phys., 42,4636 (1971).
Dynamics of Stretched Polymer Chains P. Pincus* Physique de la Matiere Condensee, College de France, 75232 Paris Cedex, France, and Laboratoire de Physique des Solides, Uniuersite Paris-Sud, 91405 Orsay, France. Received January 21, 1976
z
ABSTRACT We consider the internal modes of a strongly stretched chain ( ~ / R ,>>I 1, is the average end-to-end length, and Ro is the radius of the free coil) including both excluded volume effects and hydrodynamic interactions. It is shown that the tensile screening length tt = (@f)-’for excluded volume effects’ also plays a similar role for the hydrodynamic interactions.2Scaling arguments are then employed to derive expressions for the width of the quasielastic incoherent neutron scattering peak. All our results are restricted to dimensional p o w e r laws and lack precise numerical coefficients.
I. Introduction The purpose of this investigation is to study the internal dynamics of deformed, isolated, flexible polymers. The deformation is achieved by an external tensile force f applied to the ends of the chain. Such a situation might obtain, for example, (a) with polar molecules in electric fields (assuming * John Simon Guggenheim Fellow. Address correspondence to the Depart ment of Physics, University of California, Los Angeles, California 90024.
that the monomer moment has a component along the backbone axis), (b) in the presence of strong velocity gradients3 (although the nature of the force is slightly more complex in this case), (c) for a chain portion between cross-links in a stretched network. In a previous paper,’ we demonstrated that a long, stretched, flexible polymer in the presence of excluded volume interaction could be considered as an ideal coil of “tensile blobs” of radius Ft = (pf)-’ [/3 = ( k e T ) - l ] ;i.e., for distances
Dynamics of Stretched Polymer Chains
Vol. 10, No. 1, January-February 1977 exceeding tt the excluded volume effects are effectively screened out while within a blob they are maintained. In particular, it was determined that for it < R F( R F= aN' is the Flory radius where N is the polymerization index, a is the monomer size, and u = % in three dimensions) the stress-strain relationship became nonlinear. The scaling equation for the average end-to-end length in response to a tension f directed along the z axis is
z
-
z=
(1)
RF@"(RF/[t)
where @ ( x ) x for x > 1 where p = u - l - 1. This leads to 0: f2I3in the strongly stretched regime. The present paper deals with quasimacroscopic movements of a polymer in the strongly stretched limit ( [ t ) = 0
= ((a,>)*) = 1 a2 - -
3
45
a
/it2
(8)
The most important effect of stretching is to modify the average distance between a given pair of monomers to give 9Ft2
-
i.e., for Et >> ( a / 3 )(1 n - m I )I/*, the usual r n lI2a random walk behavior is maintained, while for [t S({Ja)'. There is an anisotropy in the relaxation spectrum generated by the deformations, i.e., there exists T~ # T ~ , However, the dependences on such physical variables as the momentum transfer remains identical. Since it is these features which interest us here, we shall not explicitly discuss this anisotropy, but it should be kept in mind when experiments are performed. In the long-wavelength, low-
212
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Macromolecules
frequency limit p > 1, we find
(14) where a is a constant of order unity. This is the Zimm5 and Dubois-Violette-de Gennes7 result and leads to (4) for the width of the quasielastic neutron peak. For pS,,, > 1, 1) Po, must be independent of the molecular weight; i.e., on the scale of the probe wavelength, the coil is effectively infinite. Assuming a power law [ F l ( x ) x p , x >> 11 this gives p = 4 for N to cancel in (21). For Et > &-I, we expect to find Aaq independent of the screening for q & length; Le., Fl(qt) 1 for qEt 1and Fl(qtt) (sit)" >> 1where r is found from the condition that Awq is dependent of Et. This leads to r = -ll? and the usual Zimm q3 form (eq 4).
-
-
-
IV. Conclusion In summary, we have shown that for stretched polymers the parameter lt = ((?f)-' in addition to being the screening length for the excluded volume effects is also the characteristic distance beyond which the hydrodynamic interactions lead to only logarithmic corrections to free draining behavior. Then, using scaling arguments, we found functional dependences for the width of the incoherent inelastic neutron scattering peak. For sufficiently short wavelengths, q & >> 1, the ideal chain q" behavior obtains, characteristic of complete hydrodynamic coupling between the monomers. For R F -
